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preprint - UTF 301

Finite temperature effective potential

on hyperbolic spacetimes

Guido Cognola1

Dipartimento di Fisica, Universita di Trento

and Istituto Nazionale di Fisica Nucleare

Gruppo Collegato di Trento, Italia

Klaus Kirsten2

Dipartimento di Fisica, Universita di Trento, Italia

Luciano Vanzo3, Sergio Zerbini4

Dipartimento di Fisica, Universita di Trento

and Istituto Nazionale di Fisica Nucleare

Gruppo Collegato di Trento, Italia

Abstract: The finite temperature one-loop effective potential for a scalar field defined on an

ultrastatic spacetime, whose spatial part is a compact hyperbolic manifold, is studied. Dif-

ferent analytic expressions, especially valuable at low and high temperature are derived.

Based on these results, the symmetry breaking and the topological mass generation are

discussed.

PACS number(s): 03.70+k, 11.10.Gh

1 Introduction

One major area of research in quantum field theory in curved spacetime in the 1980s has beenon interacting quantum fields and their implications ([1, 2], for a review see [3]). The centralobject of physical interest is the effective potential which should define the dynamics of theearly universe at scales bigger than the Planck scale. In particular in new inflationary models[4, 5] the effective cosmological constant is obtained from an effective potential which includesquantum corrections to the classical potential of a scalar field [6].

Unfortunately there are a lot of difficulties in calculating the effective potential or action in aquantum field theory in a general curved spacetime [3]. Only approximations like the derivativeexpansion in the background field at zero [7, 8, 9] and at finite temperature [10, 11] are available.

1e-mail: cognola@itnvax.science.unitn.it, cognola@itncisca.bitnet, 37953::cognola2 Address after October 1, 1993: Department E.C.M., Faculty of Physics, Barcelona University, Diagonal 647,

E-08028 Barcelona, Spain3e-mail: vanzo@itnvax.science.unitn.it, vanzo@itncisca.bitnet, 37953::vanzo4e-mail: zerbini@itnvax.science.unitn.it, zerbini@itncisca.bitnet, 37953::zerbini

1

Therefore it is very natural to deal with some specific spaces which are interesting from thecosmological viewpoint. For example the role of constant curvature [12] and of anisotropy indifferent Bianchi type universes [13, 14, 15, 16, 17, 18, 19, 20] has been considered in detail.

Furthermore the importance of nontrivial topology (imposed, for example, by finite tempera-ture or compactified spatial sections) has been emphasized at several places. Apart from its wellknow influence on the effective mass of the field [21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34]and consequently also on particle creation [35, 36], let us mention as a motivation for such con-siderations that possibly the universe as a whole exhibits nontrivial topology ([37, 38, 39] andreferences therein).

Most of the mentioned literature has been concerned with only one compact dimension(representing imaginary time or a compact spatial dimension) and with a spacetime of van-ishing curvature. However, in a cosmological setting the inclusion of curvature is of coursenecessary. For that reason, in a previous paper [40] we started recently investigations on the in-fluence of nontrivial topology together with nonvanishing curvature on the effective potential ina self-interacting scalar field theory. In that article we restricted ourselves to zero temperature.However, in the early universe the inclusion of nonvanishing temperature is necessary and thisis the aim of the following article, which is organized as follows.

In Sec. 2 we remind the reader very briefly of the definition of the effective potential in aself-interacting scalar field theory. Afterwards different techniques are applied to find analyticresults for the potential. First, in Sec. 3, the Selberg trace formula is used to derive a verycompact form of the effective potential of the theory. Then, in Sec. 4, other representations ofsuch a result, but especially useful at low and high temperature, are given. For both results thephysical implications in terms of the mass of the field are discussed in Sec. 5. The main resultsare summarized in the conclusions.

2 One-loop effective potential of the self-interacting theory

The concept of the effective action and the effective potential is well discussed in the literature[2, 41, 6, 42] and the introduction of these quantities will be very brief.

We consider a self-interacting scalar field coupled to gravity on a four-dimensional spacetimewith a compact hyperbolic spatial section H3/Γ, H3 being the Lobachevsky space and Γ adiscrete group of isometries of H3, which is assumed to have only hyperbolic elements, so theRiemannian structure is smooth. The theory at finite temperature T = 1/β may be obtained bycompactifying the imaginary time τ and assuming the field φ(τ, ~x) to fulfill periodic boundaryconditions with respect to τ , this means φ(τ, ~x) = φ(τ +β, ~x). In this way, the manifold becomesan Euclidean space of the form M = S1 × H3/Γ, S1 being a circumference.

The theory is described by the action

S(φ) =

∫ β

0dτ

∫

H3/Γ

[

−1

2φ

(

∂2

∂τ2+ ∆S

)

φ + V (φ)

]

|g| 12 d3x , (2.1)

with the Laplace-Beltrami operator ∆S acting on the spatial section H3/Γ of M and the classicalpotential

V (φ) =m2φ2

2+

ξRφ2

2+

λφ4

24. (2.2)

As usual R is the scalar curvature of M, while ξ and λ are arbitrary dimensionless parameters(coupling constants).

The action (2.1) has a minimum at φ = φ which satisfies the classical equation of motion

−(

∂2

∂τ2+ ∆S

)

φ + V ′(φ) = 0 . (2.3)

2

Quantum fluctuations φ′ = φ − φ around the classical background φ satisfy to lowest order inφ′ an equation of the form

Aφ′(x) =

(

− ∂2

∂τ2− ∆S + V ′′(φ)

)

φ′(x) =

(

− ∂2

∂τ2+ L

)

φ′(x) = 0 , (2.4)

where we have introduced the operator L = −△S + V ′′(φ).Assuming a constant background field φ the concept of the effective potential is well defined

[20]. Formally it is given by

Veff (φ) = V (φ) + hV(1)T (φ) + O(h2) , (2.5)

with the purely classical part V (φ) and with the one-loop quantum corrections

V (1)(φ) =1

2βVSln det(Aℓ2) = V

(1)0 (φ) + V

(1)T (φ) , (2.6)

where VS is the volume of the spatial hyperbolic part and ℓ a length renormalization scaling

parameter. Moreover, we distinguished between zero ( V(1)0 (φ)) and finite (V

(1)T (φ)) temperature

contributions. Using ζ-function prescription for the regularization of functional determinants[43, 44], Eq. (2.6) assumes the form

V (1)(φ) = − 1

2βVS

[

ζ ′(0|A) + ζ(0|A) ln ℓ2]

, (2.7)

where ζ(s|A) is the ζ-function associated with the operator A and ζ ′(s|A) its derivative withrespect to s. More explicitly, using the Poisson summation formula, one has

ζ(s|A) =∞∑

n=−∞

∑

j

[

ω2j + (2πn/β)2

]−s=

∞∑

n=−∞

ζ(s|L + [2πn/β]2) (2.8)

=β

π

∞∑

n=−∞

∫

∞

0ζ(s|L + t2) cos nβt dt , (2.9)

ω2j being the eigenvalues of the operator L. The term n = 0 in Eq. (2.8) leads to the function

ζ(s|L) =∑

j

(

ω2j

)−son the spatial section. In the explicit computation, it is convenient to

distinguish between n = 0 and n 6= 0. Then, using Eq. (2.9) and the Mellin representation ofζ-function, we obtain

ζ(s|A) − βΓ(s − 12)ζ(s − 1

2 |L)√πΓ(s)

=2β

π

∞∑

n=1

∫

∞

0ζ(s|L + t2) cos nβt dt , (2.10)

=β√

πΓ(s)

∞∑

n=1

∫

∞

0ts−3/2e−n2β2/4t Tr e−tL dt . (2.11)

We note that the n = 0 term in Eqs. (2.11) and (2.10) gives the zero temperature contribution

V(1)0 (φ) to the effective potential. Such a contribution, which requires renormalization, has been

extensively considered in Ref. [40] to which we refer the reader for a detailed discussion. Here weshall concentrate our attention only on the effects due to non zero temperature. Using Eqs. (2.7),(2.10) and (2.11) we get

V(1)T (φ) = − 1

πVSlims→0

∞∑

n=1

∫

∞

0ζ ′(s|L + t2) cos nβt dt , (2.12)

= − 1√4πVS

∞∑

n=1

∫

∞

0t−3/2e−n2β2/4t Tr e−tL dt , (2.13)

=1

βVSTr ln

(

1 − e−βQ)

, (2.14)

3

where the operator Q = |L|1/2 has eigenvalues ωj. As it is well known, low temperature contri-butions to the effective potential are exponentially vanishing.

In view of a high temperature expansion, it is convenient to write Eq. (2.13) in a more usefulform. To this aim we recall the Mellin transforms

f(z) =

∫

∞

0xz−1f(x) dx , (2.15)

f(x) =1

2πi

∫

Re z=cx−zf(z) dz , (2.16)

c being a real number belonging to the strip in which f(z) is analytic, and the Mellin-Parsevalidentity

∫

∞

0f(x)g(x) dx =

1

2πi

∫

Re z=cf(z)g(1 − z) dz , (2.17)

valid for any pair of functions f , g with Mellin transforms f , g. Then choosing f(t) =t−3/2 Tr e−tL, g(t) = e−n2β2/4t and using Eqs. (2.13) and (2.17), we obtain the so called Mellin-Barnes representation for the statistical sum contribution [45]

V(1)T (φ) = − 1

2πiVS

∫

Re s=cΓ(s − 1)ζR(s)ζ(s−1

2 |L) β−s ds . (2.18)

This is an exact expression, which is particularly useful in the derivation of the high temperatureexpansion of the effective potential, once the ζ-function related to the operator L is known. InEq. (2.18), ζR(s) is the usual Riemann ζ-function.

3 Selberg trace formula

The ζ-function of the Laplace-Beltrami operator acting on the compact hyperbolic manifoldH3/Γ can be related to the properties of the discrete group Γ of isometries of H3 by means ofSelberg trace formula. To state such a relation, let h(r) be an even and holomorphic function ina strip larger than 2 about the real axis and h(r) = O(r−(3+ε)) uniformly in this strip as r → ∞.Then the Selberg trace formula reads [46, 47] (here we are considering untwisted scalar fields,namely functions such that φ(γx) = φ(x), ∀γ ∈ Γ)

∞∑

j=0

h(rj) =VF

2π2

∫

∞

0r2h(r) dr +

∑

P

∞∑

k=1

lγ h(klγ)

S3(k; lγ). (3.1)

As usual, we normalize the constant curvature to κ = R/6 = −1. In this manner all quantitieswill be dimensionless. Here VF is the volume of the fundamental domain F , which, in normalunits, is just the volume of H3/Γ, ans h is the Fourier transform of h, that is

h(u) =1

2π

∫

∞

−∞

e−iruh(r) dr . (3.2)

To help the reader understanding the formula, we briefly explain the double series on the righthand side. The symbol P denotes the set of all primitive closed geodesics in the compactmanifold, namely, those geodesics passing only once through any given point. If γ ∈ P then itslength is denoted by lγ. The integer k is the winding number of any non primitive geodesic.Hence, the sum over all primitive geodesics followed by the sum over k, which appears in the traceformula, is nothing but the sum over all geodesics of the compact manifold. This is expectedbecause every geodesic must contribute to the trace. Now to every closed primitive geodesic γ,with length lγ, we can associate an isometry, also denoted γ, namely, the unique isometry such

4

that the hyperbolic distance d(x, γx) = lγ . Here x ∈ H3 is any point whose projection in thequotient manifold belongs to γ. Thus γ ∈ PSL(2, IC) (the isometry group of H3) and so it canbe conjugated to a diagonal matrix, whose upper entry we call aγ (the lower entry is then a−1

γ ).

A simple calculation will shows that lγ = ln(|aγ |) and finally S3(k; lγ) = |akγ − a−k

γ |2. This isseen to be essentially the Jacobian determinant of the parallel transport around γ (see Ref. [48]for details).

In Eq. (3.1), the sum on the left hand side is over the spectrum of the Laplace operatorwith eigenvalues −λj, and rj =

√

λj − 1, if λj > 1, while rj = i√

1 − λj if λj < 1. Finally, theintegral is the contribution of those geodesics which are contractible to a point, i.e. it is thedirect path contribution to the trace.

In order to compute Tr exp(−tL) by Selberg trace formula, we may choose h(r) = exp(−t[r2+M2]), whose Fourier-transform reads

h(u) =Me−tM2−u2/4t

√4πt

(3.3)

Here we have set

M2 = V ′′(φ) − κ = m2 +λφ2

2+

(

ξ − 1

6

)

R . (3.4)

By a simple calculation we obtain

Tr e−tL =VFe−tM2

(4πt)3/2+

e−tM2

(4πt)1/2

∑

P

∞∑

k=1

lγe−k2l2γ/4t

S3(k; lγ), (3.5)

ζ(s|L) =VF

(4π)3/2

Γ(s − 3/2)M3−2s

Γ(s)

+∑

P

∞∑

k=1

lγS3(k; lγ)

[

klγ2M

]s−1/2 Ks−1/2(Mklγ)

π1/2Γ(s), (3.6)

while the effective potential reads

V(1)T (φ) = −M4

2π2

∞∑

n=1

K2(nβM)

(nβM)2

− M2

πVS

∞∑

n=1

∑

P

∞∑

k=1

lγK1

(

M√

n2β2 + k2l2γ

)

S3(k; lγ)M√

n2β2 + k2l2γ, (3.7)

where Kν(s) represents the modified Bessel function. From Eq. (3.6) we immediately get theresidue of ζ(s|L) in the poles s = 3/2 − k (k = 0, 1, 2 . . .). That is

Res(ζ(s|L); s = 3/2 − k) =(−1)kM2k

(4π)3/2k!

VF

Γ(3/2 − k). (3.8)

To complete this section we introduce the Selberg Ξ and Z funtions, which contain alltopological informations. They are defined by means of formula

Ξ(s) =d

dslnZ(s) =

∑

P

∞∑

k=1

lγe−(s−1)klγ

S3(k; lγ), (3.9)

and can be found for example in Ref. [46] to which we ones more refer the reader for moredetails. Since the number of closed primitive geodesics with a given length lγ is asymptotically

5

l−1γ exp(2lγ) and S3(k; lγ) ≃ exp(klγ), the series defining the Ξ(s)-function is convergent forRe s > 2. The Selberg function Z(s), on the other hand, is entire of order 3 and has zeroes atthe points sj = 1 ± irj , whose order is the eigenvalues multiplicity.

By means of the above functions, it is possible to rewrite the topological contribution toζ(s|L). In fact we obtain

ζ(s|L) =VF

(4π)3/2

Γ(s − 3/2)M3−2s

Γ(s)

+sinπs M1−2s

π

∫

∞

1Ξ(1 + Mu) (u2 − 1)−s du , (3.10)

from which easily follows

ζ ′(0|L) =VFM3

6π− ln Z(M + 1) . (3.11)

Now, replacing M2 with M2 + t2 in the latter formula and using Eq. (2.12), we obtain thenice representation

V(1)T (φ) = −M4

2π2

∞∑

n=1

K2(nβM)

(nβM)2

+1

πVS

∞∑

n=1

∫

∞

0ln Z

(

1 +√

[t2 + M2]/|κ|)

cos nβt dt , (3.12)

where normal units have been restored.

4 Low and high temperature expansions

The low temperature behaviour of the theory can be directly obtained from Eq. (3.7) since theasymptotic expansion of modified Bessel function is well known. In fact we have

V(1)T (φ) ∼ − 1

2VS

M2

(2π)1/2

∞∑

n=1

∑

P

∞∑

k=1

lγe−nβM√

1+(klγ/nβ)2

S3(k; lγ)(

nβM√

1 + (klγ/nβ)2)3/2

×

1 +

3

4(

nβM√

1 + (klγ/nβ)2)

− 1

2

M4

(2π)3/2

∞∑

n=1

e−nβM

(nβM)5/2.(4.1)

We see that the leading term comes from the topological part.In order to get the high temperature expansion of the effective potential, we shall perform

the integration in Eq. (2.18). To this aim we observe that the integrand function in Eq. (2.18)has simple poles at the points s = 4, 2, 1,−2k (k = 1, 2, . . .) and a double pole at s = 0.Hence integrating this function along a path enclosing all the poles, we obtain a series expansionin powers of β, whose coefficients are related to the residues of ζ-function, which have beencomputed in Sec. 3 (see also Ref. [49]). After a straightforward computation, using Eq. (3.10)we obtain the high temperature expansion (here γ is the Euler-Mascheroni constant)

V(1)T (φ) ∼ −π2T 4

90+

M2T 2

24−[

M3

12π− M

2VSlnZ(1 + M)

]

T

− M4

32π2

[

lnM

4πT+ γ − 3

4

]

(4.2)

− M2

2πVS |κ|1/2

∫

∞

1Ξ(1 + tM |κ|−1/2)

√

t2 − 1 dt + O(T−2) ,

6

which in the case of flat, topological trivial space, is in agreement with results of Ref. [50].It is interesting to note that all coefficients of the negative powers of T are proportional to

the residues of ζ(s|L) and so they do not depend on the topology. Moreover, the topological termindependent of T in the latter formula is the same, but the sign, as the topological contributionto the zero temperature effective potential, also after renormalization (see Ref. [40]). This meansthat topology enters the high temperature expansion of the effective potential only with a termproportional to T .

5 Phase transitions of the system

The relevant quantity for analyzing the phase transitions of the system is the mass of the field.The quantum corrections to the mass are defined by means of equation

V (1)(φ) = Λeff +1

2(m2

0 + m2T )φ2 + O(φ4) , (5.1)

where Λeff (the cosmological constant) in general represents a complicated expression not de-

pending on the background field φ while m0 and mT represent the zero and finite temperaturequantum corrections to the mass m. As has been shown in Ref. [40], m0 has curvature andtopological contributions, which help to break the symmetry (for ξ < 1/6). On the contrary,here we shall see that mT always helps to restore the symmetry.

By evaluating the second derivative with respect to φ at the point φ = 0 of Eqs. (2.14) and(3.12), we obtain for mT the two equations

m2T =

λ

2VSTr

e−βQ0

Q0(1 − e−βQ0), (5.2)

=λM2

0

4π2

∞∑

n=1

K1(nβM0)

nβM0

+λ

2πVS |κ|∞∑

n=1

∫

∞

0

Ξ(1 +√

[t2 + M20 ]/|κ|)

√

[t2 + M20 ]/|κ|

cos nβt dt , (5.3)

where Q0 = Qφ=0 = | − △S + m2 + ξR|1/2 and M0 = Mφ=0 = |m2 + (ξ − 1/6)R|1/2. From theexact formula, Eq. (5.2), we see that the finite temperature quantum corrections to the massare always positive, their strength mainly depending on the smallest eigenvalues of the operatorQ0. This means that such a contribution always helps to restore the symmetry. The secondvery interesting expression, Eq. (5.3), gives the mass in terms of geometry and topology of themanifold. In fact, the Ξ-function is strictly related to the isometry group Γ, which realizes thenon trivial topology of M.

To go further, we compute the corrections to the mass in the low and high temperaturelimits. Using Eqs. (4.1) and (4.2) we obtain

m2T ∼ λ

4

M20

(2π)3/2

∞∑

n=1

e−nβM0

(nβM0)3/2

+λ

4(2π)1/2VS

∞∑

n=1

∑

P

∞∑

k=1

lγe−nβM0

√1+(klγ/nβ)2

S3(k; lγ)(

nβ√

1 + (klγ/nβ)2)1/2

(5.4)

×

1 +

1

4(

nβM0

√

1 + (klγ/nβ)2)

,

7

m2T ∼ λT 2

24− 3M0λT

8π

+|κ|−1/2

4M20 VS

[

ln Z(1 + M0|κ|−1/2) + M0|κ|−1/2 Ξ(1 + M0|κ|−1/2)]

M0λT , (5.5)

which are valid for low and high temperature respectively.

6 Conclusions

In this paper we considered a self-interacting scalar field living on an ultrastatic spacetimewhose spatial part is a compact hyperbolic manifold. We were especially interested in thefinite temperature effective potential of the theory, which serves as the effective cosmologicalconstant in new inflationary models. As we have seen, the main technical tool to obtain theeffective potential was the Selberg trace formula, Eq. (3.1). Analytical expressions for theeffective potential suitable in the low, Eq. (4.1), and high temperature limit, Eq. (4.2), havebeen obtained. Whereas at T = 0 for ξ < 1

6 quantum corrections to the classical potential canhelp to break symmetry [40], the results show that, if the temperature is high enough, quantumcorrections always help to restore the symmetry (see Eq. (5.5)) .

Acknowledgments

K. Kirsten is grateful to Prof. M. Toller, Prof. R. Ferrari and Prof. S. Stringari for the hospitalityin the Theoretical Group of the Department of Physics of the University of Trento.

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